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Problem 3 (30 points) An emergency relief agency is planning flights of small helicopters to deliver medical, food, and housing supplies for populations cutoff by a recent hurricane. The following table shows the fraction of each plane's weight w; and volume vi capacity of shipping containers for different materials to be sent, along with the number that must be transported. i Material 1 First aid supplies 2 Drinking water 3 Diesel Generators 4 Generator fuel 5 Tents 6 Cots 7 Blankets 8 Rain capes Weight Fraction wi Volume Fraction vi Quantity Needed qi 0.04 0.10 30 0.20 0.14 20 0.40 0.24 12 0.28 0.32 0.10 0.28 15 0.16 0.24 30 0.03 0.18 40 0.08 0.14 25 23 The agency wants to meet all these needs with the minimum number of flights. (a) Formulate the problem as an MIP over decision variables x; = the number of times load combination j is used, with columns for load combinations made up of aij the number of units of material i carried in each load j. (b) Discuss how the large number of feasible load mixes make the Column Generation (CG) method attractive for this application. (c) Are the weight and the volume constraints column coefficients dij required to satisfy in terms of parameters w; and v;? Explain. (d) Construct an initial set of columns j for a first Restricted Master Problem (RMP) as ones for pure loads with dij = 0 except on one product i where the maximum feasible number of that product is specified. Write the RMP for this initial set of columns. Problem 3 (30 points) An emergency relief agency is planning flights of small helicopters to deliver medical, food, and housing supplies for populations cutoff by a recent hurricane. The following table shows the fraction of each plane's weight w; and volume vi capacity of shipping containers for different materials to be sent, along with the number that must be transported. i Material 1 First aid supplies 2 Drinking water 3 Diesel Generators 4 Generator fuel 5 Tents 6 Cots 7 Blankets 8 Rain capes Weight Fraction wi Volume Fraction vi Quantity Needed qi 0.04 0.10 30 0.20 0.14 20 0.40 0.24 12 0.28 0.32 0.10 0.28 15 0.16 0.24 30 0.03 0.18 40 0.08 0.14 25 23 The agency wants to meet all these needs with the minimum number of flights. (a) Formulate the problem as an MIP over decision variables x; = the number of times load combination j is used, with columns for load combinations made up of aij the number of units of material i carried in each load j. (b) Discuss how the large number of feasible load mixes make the Column Generation (CG) method attractive for this application. (c) Are the weight and the volume constraints column coefficients dij required to satisfy in terms of parameters w; and v;? Explain. (d) Construct an initial set of columns j for a first Restricted Master Problem (RMP) as ones for pure loads with dij = 0 except on one product i where the maximum feasible number of that product is specified. Write the RMP for this initial set of columns